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How can I define a orthoplex as a polyhedron?

The definitions that I am using are:

  • $n$-dimensional orthoplex: $$O_{n} = \mbox{conv}\left\{e_1, -e_1, e_2, -e_2,\dots, e_n, -e_n \right\}$$ where $e_1,\dots,e_n$ are the vectors of the standard basis. $\mbox{conv}$ denotes a convex combination of finitely many elements in $O_{n}$.

  • polyhedron $$\{x\in\mathbb{R}^n : Ax \leq b\}$$

I tried to get an arbitrary element in the orthoplex and write it as convex combination, but I'm struggling doing this and don't know how I can write that to a matrix formula.

Rodrigo de Azevedo
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mathastic
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  • There is a lot of information about this polytope at https://en.wikipedia.org/wiki/Cross-polytope . If that does not provide what you need, [edit] the question to tell us more precisely what you want in a "matrix formula". – Ethan Bolker Dec 09 '21 at 16:12
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    Thank you for the links Ethan and Rodrigo. I do understand the question now and I also understand this subject better now – mathastic Dec 09 '21 at 16:45

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